Separability and distillability in composite quantum systems-a primer

Abstract Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like quantum gravity, string theories, etc. concern quantum theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: (a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement; and (b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system). Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of quantum information theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.

[1]  R. Jozsa,et al.  Quantum Computation and Shor's Factoring Algorithm , 1996 .

[2]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[3]  M. Lewenstein,et al.  Volume of the set of separable states , 1998, quant-ph/9804024.

[4]  C. Adami,et al.  Quantum extension of conditional probability , 1999 .

[5]  John J. Hopfield,et al.  On the existence of magnetic monopoles , 1964 .

[6]  P. Horodecki.  Separability criterion and inseparable mixed states with positive partial transposition , 1997, quant-ph/9703004.

[7]  Asher Peres,et al.  Quantum Theory: Concepts and Methods , 1994 .

[8]  Paul Busch,et al.  Remarks on separability of compound quantum systems and time reversal , 1997 .

[9]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[10]  M. Horodecki,et al.  Quantum α-entropy inequalities: independent condition for local realism? , 1996 .

[11]  M. Horodecki,et al.  Reduction criterion of separability and limits for a class of distillation protocols , 1999 .

[12]  M. Horodecki,et al.  BOUND ENTANGLEMENT CAN BE ACTIVATED , 1998, quant-ph/9806058.

[13]  Ashish V. Thapliyal,et al.  Evidence for bound entangled states with negative partial transpose , 1999, quant-ph/9910026.

[14]  A. Messiah.  Quantum Mechanics , 1961 .

[15]  G. Vidal,et al.  Robustness of entanglement , 1998, quant-ph/9806094.

[16]  M. Horodecki,et al.  Inseparable Two Spin- 1 2 Density Matrices Can Be Distilled to a Singlet Form , 1997 .

[17]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[18]  M. Planck,et al.  Über eine Verbesserung der Wienschen Spektralgleichung , 1978 .

[19]  D. Bruß,et al.  Construction of quantum states with bound entanglement , 1999, quant-ph/9911056.

[20]  M. Horodecki,et al.  Separability of mixed states: necessary and sufficient conditions , 1996, quant-ph/9605038.

[21]  G. Vidal,et al.  LOCAL DESCRIPTION OF QUANTUM INSEPARABILITY , 1998 .

[22]  N. Gisin.  Hidden quantum nonlocality revealed by local filters , 1996 .

[23]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[24]  Deutsch,et al.  Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels. , 1996, Physical review letters.

[25]  M. Lewenstein,et al.  Separability and entanglement of composite quantum systems , 1997, quant-ph/9707043.

[26]  M. Horodecki,et al.  Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature? , 1998, quant-ph/9801069.